A symplectic Lie groupoid is a Lie groupoid with a
multiplicative symplectic form. We take the perspective that such an object is symplectic manifold with an extra categorical structure. Applying the machinery of Floer theory, the extra structure is expected to yield a monoidal structure on the Fukaya category, and new operations on the closed string invariants. I will take an examples-based approach to working out what these structures are, focusing on cases where the
Floer theory is tractable, such as the cotangent bundle of a compact manifold.

Hilbert's 12th problem is to provide explicit analytic formulae for elements generating the maximal abelian extension of a given number field. In this talk I will describe an approach to Hilbert’s 12th that involves proving exact p-adic formulae for Gross-Stark units. This builds on prior joint work with Kakde and Ventullo in which we proved Gross’s conjectural leading term formula for Deligne-Ribet p-adic L-functions at s=0. This is joint work with Mahesh Kakde.

These two talks will introduce the asymptotic rank and asymptotic subrank of tensors and graphs - notions that are key to understanding basic questions in several fields including algebraic complexity theory, information theory and combinatorics.

Matrix rank is well-known to be multiplicative under the Kronecker product, additive under the direct sum, normalized on identity matrices and non-increasing under multiplying from the left and from the right by any matrices. In fact, matrix rank is the only real matrix parameter with these four properties.

Following a program proposed by Gromov, we study metric singularities of positive scalar curvature of codimension two and three. In addition, we describe a comparison theorem for positive scalar curvature that is captured by polyhedra. Part of this talk is based on joint work with C. Mantoulidis.

We'll describe a joint project with X. Zhou in which we use min-max techniques to prove existence of closed hypersurfaces with prescribed mean curvature in closed Riemannian manifolds. Our min-max theory handles the case of nonzero constant mean curvature, and more recently a generic class of smooth prescription functions, without assuming a sign condition.